1. Field of Invention
The present invention relates to a method for computing an exact impulse response of a plane acoustic reflector at zero offset due to a point acoustic source useful for seismological applications.
A novel and an exact algebraic formula for the impulse response of a plane acoustic reflector at zero offset due to a point acoustic source underlies the method cited in the present invention. An observation at zero offset represents the case when the seismic source and the receiver are located either at the same point or arbitrarily close to each other.
2. Background and Prior Art of the Invention
Reflection seismology is a widely employed tool for exploration of hydrocarbons. The seismic exploration industry is focused on a particular type of experiment or, more precisely, a particular type of ensemble of experiments. A source of seismic energy is set off at a shallow depth near the surface of the earth and the upward scattered signal from the subsurface is recorded at an array of receiver located on the surface, relatively near the source compared to the depth of penetration of the signal into the earth. The distance from the source to a receiver is called offset. The output of the receiver is recorded from the time the source was set off. That is, the data represents the amount of time needed for a wave to travel down to a reflector of interest and back to the recording surface. Seismic data is grouped into traces, lines and surveys. A trace is an output of a receiver and a line is a group of traces, while a survey is a collection of lines from a geographical location.
The entire array, source and receivers, is moved and the experiment is repeated many times. The interval of separation between the successive experiments will be of the order of a few tens of meters, with the full set of experiments carried out along a line, which is a few kilometers to a few tens of kilometers in length, or even an areal array whose sides are of those dimensions. In a widely used processing step, data for coincident source and receiver is synthesized from the small offset data of the ensemble of experiments. This is achieved by time-shifting the data of each small offset experiment to approximate the arrival time that would occur, if the experiment had been a backscatter experiment at the midpoint between the source and the receiver, simulating a zero-offset condition. There are a number of positive consequences of this synthesis. For example, there would be many experiments in the ensemble whose sources and receivers will share a common midpoint. By adding (stacking) all of these together with proper time-shift for each of them, one accomplishes a noise reduction. The process is called common midpoint stacking.
The information appearing on a single seismic trace does not allow one to determine the time-spatial position of the reflecting point. Each reflection event will show up as if it occurred directly beneath the recording point. It is possible, however, to make use of the apparent dips on the seismic record section, which is the collection of all the traces in a line, to estimate the true locations of the subsurface reflections. The process which restores a reflector's true subsurface position is called migration. A related process called inversion converts seismic data to the subsurface earth properties. The purpose of inversion is to replace the time-dependence of the input data with a depth-dependence. After inversion the position of a peak on a trace indicates at what depth there is a change in the wavespeed, while the amplitude of the peak relates to the magnitude of that change.
The migration and inversion algorithms used in the seismic industry assume the earth to be an acoustic medium, as assumed in the present invention.
Reference may be made to Cagniard (1939) wherein the author obtained an algebraic formula for the response of a reflector due to a line source possessing temporal variation of a Heaviside function. One drawback of the formula is that a line source is of infinite spatial extent and does not represent real seismic sources. Another drawback is that the formula is not explicit in that the solution requires, for each elapsed time, since the onset of the source, calculation of the corresponding horizontal wave slowness, which in turn yields the value of the conforming vertical slowness required in the algebraic formula. One more drawback is that this algebraic formula, unlike the algebraic formula underlying the present invention, does not have a specific term representing the asymptotic (ray-theoretical) solution.
Reference may be made to deHoop (1960) who extended the result of Cagniard (1939) to obtain an exact solution for the reflection response of a reflector due to a point source. One drawback is that the solution, unlike the algebraic formula underlying the present invention, can be stated not as an explicit algebraic expression enabling exact computation, but rather as a single finite integral that must be evaluated by numerical means entailing approximations. Another drawback is that the solution, unlike the algebraic formula of the present invention, does not have a specific term representing the asymptotic solution.
Reference may be made to Aki and Richard (1980) who derived an exact solution for the reflection response of an acoustic reflector due to a point source. One drawback of the said derivation is that their solution, unlike the algebraic formula underlying the present invention, does not have a specific term representing the asymptotic solution. Another drawback is that the solution, unlike the algebraic formula underlying the present invention, can be stated not as an algebraic formula but rather as an infinite integral that must be evaluated by numerical means entailing approximations.
Reference may be made to Hilterman (1975) who claims to derive the zero-offset reflection response of a plane acoustic reflector. A particular drawback is that the solution is valid only for a rigid boundary corresponding to a reflection coefficient of unity. Further, the solution, unlike in the present invention, demonstrates no result for an arbitrary reflection coefficient. Also, it is not obvious as to what should be done to obtain the solution for a reflector with an arbitrary reflection coefficient. The foremost drawback is that the solution is incorrect in view of the following. According to Hilterman (1975) the complete wave equation solution corresponding to a zero offset is equal to the asymptotic solution for an acoustic plane reflector. This statement is in conflict with Claerbout (1985) and Aki and Richards (1980). According to Claerbout (1985) the complete wave equation solution is equal to the asymptotic solution only for a spherical reflector and never for a plane reflector. Further, according to Aki and Richards (1980) the reflected wavefield for a plane reflector has a longer tail in time domain than the incident wave, a result inconsistent with the solution of Hilterman (1975), according to which the complete wave equation response has the same time span as the incident wave. Therefore, the solution of Hilterman (1975) for the zero-offset reflection response of a plane acoustic reflector is incorrect.